7,078 research outputs found

    Computational complexity of the ground state energy density problem

    Get PDF
    We study the complexity of finding the ground state energy density of a local Hamiltonian on a lattice in the thermodynamic limit of infinite lattice size. We formulate this rigorously as a function problem, in which we request an estimate of the ground state energy density to some specified precision; and as an equivalent promise problem, GSED, in which we ask whether the ground state energy density is above or below specified thresholds. The ground state energy density problem is unusual, in that it concerns a single, fixed Hamiltonian in the thermodynamic limit, whose ground state energy density is just some fixed, real number. The only input to the computational problem is the precision to which to estimate this fixed real number, corresponding to the ground state energy density. Hardness of this problem for a complexity class therefore implies that the solutions to all problems in the class are encoded in this single number (analogous to Chaitin's constant in computability theory). This captures computationally the type of question most commonly encountered in condensed matter physics, which is typically concerned with the physical properties of a single Hamiltonian in the thermodynamic limit. We show that for classical, translationally invariant, nearest neighbour Hamiltonians on a 2D square lattice, PNEEXP†EXPGSED† EXPNEXP, and for quantum Hamiltonians PNEEXP†EXPGSED† EXPQMAEXP. With some technical caveats on the oracle definitions, the EXP in some of these results can be strengthened to PSPACE. We also give analogous complexity bounds for the function version of GSED

    Uncomputability of phase diagrams

    Get PDF
    The phase diagram of a material is of central importance in describing the properties and behaviour of a condensed matter system. In this work, we prove that the task of determining the phase diagram of a many-body Hamiltonian is in general uncomputable, by explicitly constructing a continuous one-parameter family of Hamiltonians H(φ), where φ∈ R, for which this is the case. The H(φ) are translationally-invariant, with nearest-neighbour couplings on a 2D spin lattice. As well as implying uncomputablity of phase diagrams, our result also proves that undecidability can hold for a set of positive measure of a Hamiltonian’s parameter space, whereas previous results only implied undecidability on a zero measure set. This brings the spectral gap undecidability results a step closer to standard condensed matter problems, where one typically studies phase diagrams of many-body models as a function of one or more continuously varying real parameters, such as magnetic field strength or pressure

    Uncomputably complex renormalisation group flows

    Get PDF
    Renormalisation group methods are among the most important techniques for analysing the physics of many-body systems: by iterating a renormalisation group map, which coarse-grains the description of a system and generates a flow in the parameter space, physical properties of interest can be extracted. However, recent work has shown that important physical features, such as the spectral gap and phase diagram, may be impossible to determine, even in principle. Following these insights, we construct a rigorous renormalisation group map for the original undecidable many-body system that appeared in the literature, which reveals a renormalisation group flow so complex that it cannot be predicted. We prove that each step of this map is computable, and that it converges to the correct fixed points, yet the resulting flow is uncomputable. This extreme form of unpredictability for renormalisation group flows had not been shown before and goes beyond the chaotic behaviour seen previously

    A Scalable Control Design for Grid-Forming Inverters in Microgrids

    Get PDF
    Microgrids are increasingly recognized as a key technology for the integration of distributed energy resources into the power network, allowing local clusters of load and distributed energy resources to operate autonomously. However, microgrid operation brings new challenges, especially in islanded operation as frequency and voltage control are no longer provided by large rotating machines. Instead, the power converters in the microgrid must coordinate to regulate the frequency and voltage and ensure stability. We consider the problem of designing controllers to achieve these objectives. Using passivity theory to derive decentralized stability conditions for the microgrid, we propose a control design method for grid-forming inverters. For the analysis we use higher-order models for the inverters and also advanced dynamic models for the lines with an arbitrarily large number of states. By satisfying the decentralized condition formulated, plug-and-play operation can be achieved with guaranteed stability, and performance can also be improved by incorporating this condition as a constraint in corresponding optimization problems formulated. In addition, our control design can improve the power sharing properties of the microgrid compared to previous non-droop approaches. Finally, realistic simulations confirm that the controller design improves the stability and performance of the power network.ERC starting grant 67977

    Taking the self out of self-rule

    Get PDF
    Many philosophers believe that agents are self-ruled only when ruled by their (authentic) selves. Though this view is rarely argued for explicitly, one tempting line of thought suggests that self-rule is just obviously equivalent to rule by the self. However, the plausibility of this thought evaporates upon close examination of the logic of ‘self-rule’ and similar reflexives. Moreover, attempts to rescue the account by recasting it in negative terms are unpromising. In light of these problems, this paper instead proposes that agents are self-ruled only when not ruled by others. One reason for favouring this negative social view is its ability to yield plausible conclusions concerning various manipulation cases that are notoriously problematic for nonsocial accounts of self-rule. A second reason is that the account conforms with ordinary usage. It is concluded that self-rule may be best thought of as an essentially social concept
    • …
    corecore